Fewer runs than word length

The concept of runs, i.e. maximal periodicity or maximal occurrence of repetitions, coined by Iliopoulos et al. [10] when analysing Fibanacci words, has been introduced to represent in a succinct manner all occurrences of repetitions in a word. It is known that there are only O(n) many of them in a word of length n from Kolpakov and Kucherov [11] who proved it in a non-constructive manner. The first explicit bound was later on provided by Rytter [14]. Several improvements on the upper bound can be found in [15, 3, 13, 4, 7]. Kolpakov and Kucherov conjectured that this number is in fact smaller than n, which has been proved by Bannai et al. [1, 2]. Recently, Holub [9] and Fischer et. al. [8] gave tighter upper bounds reaching 22n/23. In this note we provide a proof of the result, slightly different than the short and elegant proof in [2].